Light Scattering: Theory and Applications
1
Fundamental Equation of Radiative Transfer

dI( ,  ,  )
 I( ,  ,  )  J ( ,  ,  )
d

J( ,  ,  ) 
4
2
(1)
1 2
  P( ,  ' ,    ' ) I( ,  ' ,  ' ) d ' d '
(2)
1 0
Phase Matrix P and Scattering Matrix F

Phase matrix usually known with reference to scattering plane: scattering
matrix F

Many different planes of scattering in multiple scattering problems

Necessary to choose one common plane of reference for Stokes parameters

Normally use local meridian plane, specified by local normal and direction of
emergence

P related to F through a coordinate transformation (from the scattering plane
to the local meridian plane)

F, in general, a function of all the angles

Special cases:
o randomly oriented particles, each with a plane of symmetry
o randomly oriented asymmetric particles and their mirror images in equal
numbers
o Rayleigh and Mie scattering

For special cases, F is a function only of scattering angle  and has the form
 F11
F
 12
0

0

F12
0
F22
0
0
F33
0
 F34
0 
0 

F34 

F44 
Isotropic Rayleigh scattering:
F11  F22 
3
(1  cos 2 )
4
F33  F44 
3
cos 
2
3
F12  F21   sin 2 
4

Mie scattering:
F11  F22 , F33  F44
3
Scattering Behavior in Different Regimes [Fig. 5, Hansen and Travis]

Rayleigh scattering
o strong positive linear polarization (  F21 / F11 ), with a maximum at 90°
scattering angle

Geometric optics
o small scattering angles: phase function large and linear polarization small
because diffracted light is predominant
o other than diffraction, most of the light scattered into forward hemisphere
due to rays passing through particle with two refractions => negative
linear polarization (Fresnel’s equations)
o positive linear polarization at ~ 80-120° scattering angle: reflection from
outside of particles
o positive polarization maximum at ~ 150° scattering angle (primary
rainbow): internal reflection – scattering angle has a maximum as a
function of the incident angle on the particle; weaker feature at ~ 120°
scattering angle (secondary rainbow): two internal reflections – scattering
angle has a minimum as a function of the incident angle on the particle
o maximum in backscattering direction (glory): incident edge rays

Transition region in between
o linear polarization complicated function of size parameter
o as size parameter decreases, degree to which paths of separate light rays
can be localized decreases
o secondary rainbow, with a more detailed ray path, lost from polarization
before primary rainbow
o primary rainbow becomes blurred with decreasing size parameter
4
Effect of Nonsphericity [Figure 1, Mishchenko et al.]
Define   F11 ( spheroid ) / F11 ( sphere) . There are five distinct regions:

Nearly direct forward scattering
o ρ~1
o region least sensitive to particle sphericity because diffraction dominates
and is determined by the average area of the particle geometrical cross
section

~10 to 30° scattering angle
o ρ>1
o ratio increases with increasing aspect ratio ε

~30 to 90° scattering angle
o ρ<1
o region becomes narrower with increasing ε
o ratio decreases with increasing ε

~90 to 150° scattering angle
o ρ >> 1
o strongly enhanced side scattering as opposed to deep and wide side
scattering minimum in spherical particles
o region becomes wider with increasing ε

~150 to 180° scattering angle
o ρ << 1
o strong rainbow and glory features suppressed by nonsphericity
In general, the polarization generated by spheroids is more neutral than that for
spheres and shows less variability with size parameter and scattering angle
[Figure 10.6, Mishchenko et al.].

>~60° scattering angle
o degree of linear polarization p is strongly ε-dependent
o deviation from spherical behavior becomes more pronounced with
increasing ε
o Lorenz-Mie theory inappropriate for nonspherical particles in this region

<~60° scattering angle
o p weakly dependent on particle shape

~120° scattering angle
o most prominent polarization feature for spheroids: bridge of positive
polarization, which separates two regions of negative or neutral
polarization at small and large scattering angles
o bridge absent in spherical particles
5
Global Climatology of Aerosol Types [Figure 2]

Seven basic aerosol types: sulfate(land/water), seasalt, carbonaceous, black
carbon, mineral dust (accumulated/coarse)

Each mixing group is a combination of 4 aerosol components

Lognormal distribution
6
Phase Function for Kahn Mixing Group Types
7
Linear Polarization for Kahn Mixing Group Types
8
Radiative Effect I: Weighting Functions in 0.76 µm O2 A Band
8
Normalized Jacobian
6
4
2
0
-2
Type
1a
1b
1c
2a
2b
3a
3b
4a
4b
4c
5a
5b
5c
-4
-6
-8
-10
0.758
0.760
0.762
0.764
0.766
0.768
0.770
0.772
Wavelength [m]
9
Radiative Effect II: Weighting Functions in 1.61 µm CO2 Band
Normalized Jacobian
0.4
Type
1a
1b
1c
2a
2b
3a
3b
4a
4b
4c
5a
5b
5c
0.2
0.0
-0.2
-0.4
1.590
1.595
1.600
1.605
1.610
Wavelength [m]
1.615
1.620
10
Radiative Effect III: Weighting Functions in 2.06 µm CO2 Band
Type
Normalized Jacobian
0.10
0.05
0.00
1a
1b
1c
2a
2b
3a
3b
4a
4b
4c
5a
5b
5c
-0.05
-0.10
2.08
2.07
2.06
2.05
2.04
Wavelength [m]
Reducing Mixing Group Types Using SSA and Extinction Behavior
Single Scattering Albedo
11
1.00
0.95
0.90
Type
1a
1b
1c
2a
4a
0.85
Group 1
0.80
Single Scattering Albedo
0.755
0.785
1.56
1.65
2.03
2.09
1.00
1.00
0.95
0.95
0.90
0.90
Type
Type
2b
4b
4c
0.85
0.80
0.755
0.785
1.56
1.65
2.03
2.09
0.85
3a
5a
5c
Group 2
0.755
Group 3
0.80
0.785
1.56
1.65
2.03
2.09
1.00
Single Scattering Albedo
1.00
Type
Group 4
5b
0.95
Type
Group 5
3b
0.95
0.90
0.90
0.85
0.85
0.80
0.80
0.755
0.785
1.56
1.65
2.03
2.09
0.755
Wavelength [m]
0.785
1.56
1.65
2.03
2.09
Normalized
Extinction Coefficient
1.1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Group 1
Type
1a
1b
1c
2a
4a
Normalized
Extinction Coefficient
0.755
1.1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Normalized
Extinction Coefficient
1.56
1.65
2.03
2.09
Group 3
Group 2
Type
Type
2b
4b
4c
0.755
1.1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.785
3a
5a
5c
0.785
1.56
1.65
2.03
2.09
0.755
0.785
1.56
1.65
2.03
1.1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
2.09
1.1
Group 5 1.0
Group 4
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Type
Type
3b
5b
0.755
0.785
1.56
1.65
2.03
2.09
0.755
Wavelength [m]
0.785
1.56
1.65
2.03
2.09
Figure 2. Global Climatology of Aerosols (Kahn et al., 2001)